the order generated by implications{0,1}, to fuzzy logic, where the truth values belong to the unit...

IntroductionNotations, Definitions and A Review of Previous Results

Main PaperReferences

The Order Generated by Implications

M. Nesibe Kesicioglu1 and Radko Mesiar2

1Recep Tayyip Erdogan University, Rize, Turkey2Slovak Technology University, Bratislava, Slovakia

FSTA, 2014

Kesicioglu and Mesiar The Order Generated by Implications



Outline

1 Introduction

2 Notations, Definitions and A Review of Previous Results

3 Main Paper

4 References




Fuzzy implications are one of the most important operationsin fuzzy logic having a significant role in many applications,viz., approximate reasoning, fuzzy control, fuzzy imageprocessing, etc.

They generalize the classical implication, which takes values in{0, 1}, to fuzzy logic, where the truth values belong to theunit interval [0, 1]. In general situation, since [0, 1] is abounded lattice, like in the case of other logical operators, theproblem of introducing implications on a bounded lattice laidbare and Ma and Wu, Logical operators on complete lattices,have introduced them at first. Several authors haveinvestigated the implications on a bounded lattice and theirrelations to the other logical operators [11, 16, 17, 20, 21, 22].




In this paper:

We introduce an order by means of an implication possessingsome special properties on a lattice and discuss some of itsproperties.

We determine the relationship between the order induced byan implication and the order on the lattice. Giving example,we show that a bounded lattice needs not be a lattice withrespect to the order induced by an implication.

Also, we give an example for an implication making the unitinterval [0, 1] a lattice with respect to the order induced by it.




Moreover, we obtain that such a generating method of anorder is independent from the order induced by an adjointt-norm (T -partial order)[10].

We prove that under the conditions required to defineimplication based order, the considered implication must bean S-implication, and so we obtain that the order induced byan implication coincides with the order which is generated in asimilar way from a t-conorm.

Consequently, we obtain that an implication on the unitinterval [0, 1] is continuous if and only if the implication basedorder and the dual of the natural order on [0, 1] coincide.




T -norm and T -conorm

Definition ( De Baets and Mesiar, 1999 )

Let (L,≤, 0, 1) be a bounded lattice. A binary operation T (S) onL is called a t-norm (t-conorm) if it satisfies the followingconditions:(1) T (T (a, b), c) = T (a, T (b, c)) (associative law),(2) T (a, b) = T (b, a) (commutative law),(3) b ≤ c ⇒ T (a, b) ≤ T (a, c) (monotonicity),(4) T (a, 1) = a (S(a, 0) = a) (boundary condition),where a, b and c are any elements of L.




The four basic t-norms on [0, 1] are the minimum TM , whichis the largest t-norm, the product TP , the Lukasiewicz t-normTL and the drastic product TD, which is the smallest t-norm,given by, respectively,TM (x, y) = min(x, y),TP (x, y) = xy,TL(x, y) = max(0, x + y − 1) and

TD(x, y) =

x if y = 1,y if x = 1,0 otherwise.

Also, t-norms on a bounded lattice (L,≤, 0, 1) are defined insimilar way, and then extremal t-norms TD as well as T∧ on Lare defined similarly as TD and TM on [0, 1].




Negation

Definition (Ma and Wu, 1991)

Let (L,≤, 0, 1) be a bounded lattice. A decreasing functionN : L → L is called a negation if N(0) = 1 and N(1) = 0. Anegation N on L is called strong if it is an involution, i.e.,N(N(x)) = x, for all x ∈ L.




On each bounded lattice L we have two extremal negationsN+, N− : L → L given by

N−(x) ={

1 if x = 0,0 otherwise

and

N+(x) ={

0 if x = 1,1 otherwise.

Obviously, for any negation N : L → L, it holds N− ≤ N ≤ N+.




Implication

Definition ( Baczynski and Jayaram, Fuzzy implications, 2008 )

Let (L,≤, 0, 1) be a bounded lattice. A binary operatorI : L2 → L is said to be an implication function, shortly animplication, if it satisfies

(I1) For every elements a, b with a ≤ b, I(a, y) ≥ I(b, y) for ally ∈ L.(I2) For every elements a, b with a ≤ b, I(x, a) ≤ I(x, b) for allx ∈ L.(I3) I(1, 1) = 1, I(0, 0) = 1 and I(1, 0) = 0.




Implication


Let (L,≤, 0, 1) be a bounded lattice. A binary operatorI : L2 → L is said to be an implication function, shortly animplication, if it satisfies(I1) For every elements a, b with a ≤ b, I(a, y) ≥ I(b, y) for ally ∈ L.

(I2) For every elements a, b with a ≤ b, I(x, a) ≤ I(x, b) for allx ∈ L.(I3) I(1, 1) = 1, I(0, 0) = 1 and I(1, 0) = 0.




Implication


Let (L,≤, 0, 1) be a bounded lattice. A binary operatorI : L2 → L is said to be an implication function, shortly animplication, if it satisfies(I1) For every elements a, b with a ≤ b, I(a, y) ≥ I(b, y) for ally ∈ L.(I2) For every elements a, b with a ≤ b, I(x, a) ≤ I(x, b) for allx ∈ L.

(I3) I(1, 1) = 1, I(0, 0) = 1 and I(1, 0) = 0.




Implication


Let (L,≤, 0, 1) be a bounded lattice. A binary operatorI : L2 → L is said to be an implication function, shortly animplication, if it satisfies(I1) For every elements a, b with a ≤ b, I(a, y) ≥ I(b, y) for ally ∈ L.(I2) For every elements a, b with a ≤ b, I(x, a) ≤ I(x, b) for allx ∈ L.(I3) I(1, 1) = 1, I(0, 0) = 1 and I(1, 0) = 0.




Note that from the definition, it follows thatI(0, x) = 1 and I(x, 1) = 1, for all x ∈ L.

Special interesting properties for implications are:

The exchange principle (EP)I(x, I(y, z)) = I(y, I(x, z)) for all x, y, z ∈ L

The left neutrality principle (NP)I(1, y) = y, for every y ∈ L

The contrapositive symmetry to a negation N (CP-N)I(x, y) = I(N(y), N(x)), for every x, y ∈ L

The left contrapositive symmetry to a negation N (L-CP(N))I(N(x), y) = I(N(y), x), for every x, y ∈ L

Obviously, for a strong negation N , the left contrapositivesymmetry and the contrapositive symmetry coincide.




Natural negation


Let (L,≤, 0, 1) be a lattice and I be an implication on L. Thefunction NI : L → L given by

NI = I(x, 0) for all x ∈ Lis a negation and it is called the natural negation of I.




S-implication

Definition (F. Karacal, 2006)

Let (L,≤, 0, 1) be a lattice. An implication I : L2 → L is called anS-implication if there exists a t-conorm S and a strong negation Nsuch that for every x, y ∈ L

I(x, y) = S(N(x), y).




T-partial order

Definition (Karacal and Kesicioglu, 2011)

Let L be a bounded lattice, T be a t-norm on L. The orderdefined as following is called a T- partial order (triangular order)for t-norm T

x �T y :⇔ T (`, y) = x for some ` ∈ L.

Proposition (Karacal and Kesicioglu, 2011)

Let T be a t-norm on a bounded lattice (L,≤, 0, 1). Then, ifx �T y necessarily we have also x ≤ y.




Definition (Klement, Mesiar, Pap, Triangular Norms, 2000)

Let T : [0, 1]2 → [0, 1] be a left-continuous t-norm. The functionIT : [0, 1]2 → [0, 1] given by

IT (x, y) = sup{z ∈ [0, 1]|T (x, z) ≤ y} (1)

is an implication and it is called as a residual implication.

Observe that the definition (1) can be applied to any t-normT : L2 → L acting on a complete lattice L, and the resultingfunction IT : L2 → L is an implication on L.




Definition

Let (L,≤, 0, 1) be a bounded lattice and I : L2 → L be animplication. Define the relation �I on L as follows:For every x, y ∈ L

y �I x :⇔ ∃` ∈ L such that I(`, x) = y. (2)




Proposition

The relation �I is a partial order on L, whenever I : L2 → L is animplication satisfying the exchange principle (EP) and thecontrapositive symmetry (CP) with respect to the strong naturalnegation NI .

We will call such an order defined in (2) as the ordering based onthe implication I.




Observe that the converse of Proposition does not hold.

Example

Consider Goguen implication I : [0, 1]2 → [0, 1] given by

I(x, y) ={

1 if x ≤ y,y/x otherwise.

Obviously, its natural negation is the Godel negation,

N−(x) ={

1 if x = 0,0 otherwise,

which is not involutive. Therefore, I can not satisfy thecontrapositive symmetry.




On the other side,

�I= {(x, y)|0 < y ≤ x ≤ 1} ∪ {(0, 0), (1, 0)} (3)

is a partial order on [0, 1], whose Hasse diagram is depicted onFigure 1.

Figure: Hasse diagram of �I given by (3)




Proposition

Let (L,≤, 0, 1) be a bounded lattice and I : L2 → L be animplication satisfying (EP) and (CP) with respect to the strongnatural negation NI . If (x, y) ∈�I , then (y, x) ∈≤.




Remark

Let (L,≤, 0, 1) be a bounded lattice and I be an implicationsatisfying (EP) and (CP-NI).

It is clear that 0 and 1 are the greatest and the least elementwith respect to �I , respectively.

The converse of the previous Proposition may not be satisfied.




For example: Consider the lattice (L = {0, a, b, c, 1},≤, 0, 1)whose lattice diagram is displayed in Figure 2:

Figure: (L = {0, a, b, c, 1},≤, 0, 1)




Define the function I : L2 → L as follows:

I 0 a b c 10 1 1 1 1 1a a 1 1 1 1b c 1 1 1 1c b 1 1 1 11 0 a b c 1

Table: The implication I on L

Obviously, I is an implication on L satisfying the exchangeprinciple (EP) and the contrapositive symmetry (CP) with respectto the strong natural negation NI defined as




NI(x) =

a if x = a,c if x = b,b if x = c,1 if x = 0,0 if x = 1.

It is clear that b ≤ c, but c �I b. The order �I on L has its Hassediagram as follows:

Figure: (L = {0, a, b, c, 1},�I , 0, 1)




Even if (L,≤, 0, 1) is a chain, the partially ordered set (L,�I)may not be a chain.

For example: consider L = [0, 1] and take the Fodorimplication I = IFD defined as

IFD(x, y) ={

1 if x ≤ y,max(1− x, y) if x > y.

(4)

It is clear that IFD satisfies the exchange principle (EP) andthe contrapositive symmetry (CP) with respect to the strongnatural negation NIFD

= NC , NC(x) = 1− x.Obviously, 1/2 and 3/4 are not comparable with respect to�IFD

.




Even if (L,≤, 0, 1) is a chain, the partially ordered set (L,�I)may not be a chain.For example: consider L = [0, 1] and take the Fodorimplication I = IFD defined as

IFD(x, y) ={

1 if x ≤ y,max(1− x, y) if x > y.

(4)

It is clear that IFD satisfies the exchange principle (EP) andthe contrapositive symmetry (CP) with respect to the strongnatural negation NIFD

= NC , NC(x) = 1− x.Obviously, 1/2 and 3/4 are not comparable with respect to�IFD

.




Remark

Let T be a left continuous t-norm on [0, 1] and IT be thecorresponding residual implication. Then, the implication basedordering and the T -partial order are independent. For example:consider the nilpotent minimum t-norm TnM given by

TnM (x, y) ={

0 x + y ≤ 1,min(x, y) otherwise.

Then, its corresponding residual implication is the Fodorimplication IFD, see (4).It is clear that 1/2 �IFD

1/8, but 1/8 �T nM 1/2 and conversely,1/2 �T nM 3/4, but 3/4 �IFD

1/2.




Notations

Let (L,≤, 0, 1) be a bounded lattice and I : L2 → L be animplication satisfying the exchange principle (EP) and thecontrapositive symmetry (CP) with respect to the strong naturalnegation NI . For X ⊆ L, we denote the set of the upper (lower)bounds of X w.r.t. �I on L by XI (XI). We denote the leastupper bound (the greatest lower bound) w.r.t. �I by ∨I (∧I).




L needs not be a lattice w.r.t. �I . The following exampleillustrates this case.

Example

Let L = [0, 1] and take the implication IFD given by (4).

For any incomparable elements x, y ∈ [0, 1], sincex ∧IFD

y = 1, (L,�IFD) is a meet-semi lattice.

There does not exist the least element of the upper bound{1/2, 3/4}IFD

= [0, 1/4).So, (L,�IFD

) is not a lattice.





Example





= [0, 1/4).

So, (L,�IFD) is not a lattice.





Example





= [0, 1/4).So, (L,�IFD

) is not a lattice.




Proposition

For every implication I satisfying (EP) and the contrapositivesymmetry (CP) with respect to the natural strong negation NI ,there exists a t-conorm S such that

I(x, y) = S(NI(x), y),

that is, I is an S-implication.




Corollary

Let I : L2 → L be an implication satisfying (EP) and thecontrapositive symmetry (CP) with respect to the natural strongnegation NI . Then, for any a, b ∈ L

a �I b if and only if NI(a) �T NI(b),where T : L2 → L is a t-norm given by T (x, y) = NI(I(x,NI(y))).




Theorem

Let I : [0, 1]2 → [0, 1] be a fuzzy implication satisfying (EP) andthe contrapositive symmetry (CP) with respect to the naturalstrong negation NI and �I be the order linked to the implicationI. Then, I is continuous if and only if �I=≥.




Proposition (Baczynski, 2010)

Let (L,≤, 0, 1) be a bounded lattice and I : L2 → L a functiondefined as

I(x, y) = N(x) ∨ y, ∀x, y ∈ L, (5)

where N : L → L is a strong negation on L. Then, I is animplication on L satisfying the exchange principle (EP) and thestrong negation N is its natural negation. Moreover, I satisfies thecontrapositive symmetry (CP) w.r.t. the natural negation N .




Proposition

Let (L,≤, 0, 1) be a bounded lattice and let I be defined as (5).Then, the order obtained from the implication I is equal to thedual of the order on L, that is, �I=≥.




Remark

For a general bounded lattice, a strong negation on L need not beexisting (see [11], example 2). If (L,≤,∧,∨) is a Boolean algebra,it can be found always a strong negation on L defined asN(x) = x′. So, the previous Proposition is always true for Booleanalgebras.




One can wonder whether L is a bounded lattice w.r.t. an orderobtained from an implication (under which conditions). In the nextProposition, we give some sufficient conditions.

Proposition

Let (L,≤, 0, 1) be a bounded lattice and I : L2 → L animplication on L defined as I(x, y) = 1 when x 6= 1 and y 6= 0,satisfying the exchange principle (EP) and the contrapositivesymmetry (CP) with respect to the strong natural negation NI ,that is the implication I : L2 → L determined by

I(x, y) =

y x = 1,NI(x) y = 0,1 otherwise.

Then, (L,�I) is a lattice.




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